2010/08/19 by L. Golinskii, Leonid Golinskiĭ, A. Kheifets +9
Computer Science · Materials Science · Mathematics · #30E05 #47A57 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Lanthanide and Transition Metal Complexes #Matrix Theory and Algorithms #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.CA #math.SP #msc:30E05 #msc:47A57
paper · pdf · doi:10.48550/arxiv.1008.3284
29 pages, substantially revised version of arXiv:0807.4017v1 [math.SP]
arxiv created 2010/08/19 · openalex publication_date 2010/08/19 · arxiv updated 2010/08/20 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28
We develop a scattering theory for CMV matrices, similar to the Faddeev--Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for the uniqueness, which are connected with the Helson--Szeg\H o and the Strong Szeg\H o theorems. The first condition is given in terms of the boundedness of a transformation operator associated to the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblunsky parameters of the CMV matrices, corresponding spectral measures and scattering functions.